Question: A circle has a sector with area $\dfrac{32}{3}\pi$ and central angle $240^\circ$. What is the area of the circle? ${16\pi}$ $\color{#9D38BD}{240^\circ}$ ${\dfrac{32}{3}\pi}$
Answer: The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{240^\circ}{360^\circ} = \dfrac{32}{3}\pi \div A_c$ $\dfrac{2}{3} = \dfrac{32}{3}\pi \div A_c$ $A_c \times \dfrac{2}{3} = \dfrac{32}{3}\pi$ $A_c = \dfrac{32}{3}\pi \times \dfrac{3}{2}$ $A_c = 16\pi$